BSP-fields: An Exact Representation of Polygonal Objects by Differentiable Scalar Fields Based on Binary Space Partitioning

The problem considered in this work is to find a dimension independent algorithm for the generation of signed scalar fields exactly representing polygonal objects and satisfying the following requirements: the defining real function takes zero value exactly at the polygonal object boundary; no extra zero-value isosurfaces should be generated; C1 continuity of the function in the entire domain. The proposed algorithms are based on the binary space partitioning (BSP) of the object by the planes passing through the polygonal faces and are independent of the object genus, the number of disjoint components, and holes in the initial polygonal mesh. Several extensions to the basic algorithm are proposed to satisfy the selected optimization criteria. The generated BSP-fields allow for applying techniques of the function-based modeling to already existing legacy objects from CAD and computer animation areas, which is illustrated by several examples

QuickCSG: Fast Arbitrary Boolean Combinations of N Solids

QuickCSG computes the result for general N-polyhedron boolean expressions without an intermediate tree of solids. We propose a vertex-centric view of the problem, which simplifies the identification of final geometric contributions, and facilitates its spatial decomposition. The problem is then cast in a single KD-tree exploration, geared toward the result by early pruning of any region of space not contributing to the final surface. We assume strong regularity properties on the input meshes and that they are in general position. This simplifying assumption, in combination with our vertex-centric approach, improves the speed of the approach. Complemented with a task-stealing parallelization, the algorithm achieves breakthrough performance, one to two orders of magnitude speedups with respect to state-of-the-art CPU algorithms, on boolean operations over two to dozens of polyhedra. The algorithm also outperforms GPU implementations with approximate discretizations, while producing an output without redundant facets. Despite the restrictive assumptions on the input, we show the usefulness of QuickCSG for applications with large CSG problems and strong temporal constraints, e.g. modeling for 3D printers, reconstruction from visual hulls and collision detection

A Method of Rendering CSG-Type Solids Using a Hybrid of Conventional Rendering Methods and Ray Tracing Techniques

This thesis describes a fast, efficient and innovative algorithm for producing shaded, still images of complex objects, built using constructive solid geometry ( CSG ) techniques. The algorithm uses a hybrid of conventional rendering methods and ray tracing techniques. A description of existing modelling and rendering methods is given in chapters 1, 2 and 3, with emphasis on the data structures and rendering techniques selected for incorporation in the hybrid method. Chapter 4 gives a general description of the hybrid method. This method processes data in the screen coordinate system and generates images in scan-line order. Scan lines are divided into spans (or segments) using the bounding rectangles of primitives calculated in screen coordinates. Conventional rendering methods and ray tracing techniques are used interchangeably along each scan-line. The method used is detennined by the number of primitives associated with a particular span. Conventional rendering methods are used when only one primitive is associated with a span, ray tracing techniques are used for hidden surface removal when two or more primitives are involved. In the latter case each pixel in the span is evaluated by accessing the polygon that is visible within each primitive associated with the span. The depth values (i. e. z-coordinates derived from the 3-dimensional definition) of the polygons involved are deduced for the pixel's position using linear interpolation. These values are used to determine the visible polygon. The CSG tree is accessed from the bottom upwards via an ordered index that enables the 'visible' primitives on any particular scan-line to be efficiently located. Within each primitive an ordered path through the data structure provides the polygons potentially visible on a particular scan-line. Lists of the active primitives and paths to potentially visible polygons are maintained throughout the rendering step and enable span coherence and scan-line coherence to be fully utilised. The results of tests with a range of typical objects and scenes are provided in chapter 5. These results show that the hybrid algorithm is significantly faster than full ray tracing algorithms

QuickCSG: Fast Arbitrary Boolean Combinations of N Solids

QuickCSG computes the result for general N-polyhedron boolean expressions without an intermediate tree of solids. We propose a vertex-centric view of the problem, which simplifies the identification of final geometric contributions, and facilitates its spatial decomposition. The problem is then cast in a single KD-tree exploration, geared toward the result by early pruning of any region of space not contributing to the final surface. We assume strong regularity properties on the input meshes and that they are in general position. This simplifying assumption, in combination with our vertex-centric approach, improves the speed of the approach. Complemented with a task-stealing parallelization, the algorithm achieves breakthrough performance, one to two orders of magnitude speedups with respect to state-of-the-art CPU algorithms, on boolean operations over two to dozens of polyhedra. The algorithm also outperforms GPU implementations with approximate discretizations, while producing an output without redundant facets. Despite the restrictive assumptions on the input, we show the usefulness of QuickCSG for applications with large CSG problems and strong temporal constraints, e.g. modeling for 3D printers, reconstruction from visual hulls and collision detection

A Survey of Methods for Converting Unstructured Data to CSG Models

The goal of this document is to survey existing methods for recovering CSG representations from unstructured data such as 3D point-clouds or polygon meshes. We review and discuss related topics such as the segmentation and fitting of the input data. We cover techniques from solid modeling and CAD for polyhedron to CSG and B-rep to CSG conversion. We look at approaches coming from program synthesis, evolutionary techniques (such as genetic programming or genetic algorithm), and deep learning methods. Finally, we conclude with a discussion of techniques for the generation of computer programs representing solids (not just CSG models) and higher-level representations (such as, for example, the ones based on sketch and extrusion or feature based operations).Comment: 29 page

Fast Exact Booleans for Iterated CSG using Octree-Embedded BSPs

We present octree-embedded BSPs, a volumetric mesh data structure suited for performing a sequence of Boolean operations (iterated CSG) efficiently. At its core, our data structure leverages a plane-based geometry representation and integer arithmetics to guarantee unconditionally robust operations. These typically present considerable performance challenges which we overcome by using custom-tailored fixed-precision operations and an efficient algorithm for cutting a convex mesh against a plane. Consequently, BSP Booleans and mesh extraction are formulated in terms of mesh cutting. The octree is used as a global acceleration structure to keep modifications local and bound the BSP complexity. With our optimizations, we can perform up to 2.5 million mesh-plane cuts per second on a single core, which creates roughly 40-50 million output BSP nodes for CSG. We demonstrate our system in two iterated CSG settings: sweep volumes and a milling simulation

Fast Reliable Ray-tracing of Procedurally Defined Implicit Surfaces Using Revised Affine Arithmetic

Fast and reliable rendering of implicit surfaces is an important area in the field of implicit modelling. Direct rendering, namely ray-tracing, is shown to be a suitable technique for obtaining good-quality visualisations of implicit surfaces. We present a technique for reliable ray-tracing of arbitrary procedurally defined implicit surfaces by using a modification of Affine Arithmetic called Revised Affine Arithmetic. A wide range of procedurally defined implicit objects can be rendered using this technique including polynomial surfaces, constructive solids, pseudo-random objects, procedurally defined microstructures, and others. We compare our technique with other reliable techniques based on Interval and Affine Arithmetic to show that our technique provides the fastest, while still reliable, ray-surface intersections and ray-tracing. We also suggest possible modifications for the GPU implementation of this technique for real-time rendering of relatively simple implicit models and for near real-time for complex implicit models

Mesh generation by domain bisection

The research reported in this dissertation was undertaken to investigate efficient computational methods of automatically generating three dimensional unstructured tetrahedral meshes. The work on two dimensional triangular unstructured grid generation by Lewis and Robinson [LeR76] is first examined, in which a recursive bisection technique of computational order nlog(n) was implemented. This technique is then extended to incorporate new methods of geometry input and the automatic handling of multiconnected regions. The method of two dimensional recursive mesh bisection is then further modified to incorporate an improved strategy for the selection of bisections. This enables an automatic nodal placement technique to be implemented in conjunction with the grid generator. The dissertation then investigates methods of generating triangular grids over parametric surfaces. This includes a new definition of surface Delaunay triangulation with the extension of grid improvement techniques to surfaces. Based on the assumption that all surface grids of objects form polyhedral domains, a three dimensional mesh generation technique is derived. This technique is a hybrid of recursive domain bisection coupled with a min-max heuristic triangulation algorithm. This is done to achieve a computationlly efficient and reliable algorithm coupled with a fast nodal placement technique. The algorithm generates three dimensional unstructured tetrahedral grids over polyhedral domains with multi-connected regions in an average computational order of less than nlog(n)